Two-loop scale-invariant scalar potential and quantum effective ...

CERN-PH-TH-2016-186

Two-loop scale-invariant scalar potential

arXiv:1608.05336v1 [hep-th] 18 Aug 2016

and quantum effective operators D. M. Ghilencea a,b , a b

Z. Lalak c and P. Olszewski c

∗

Theory Division, CERN, 1211 Geneva 23, Switzerland

Theoretical Physics Department, National Institute of Physics

and Nuclear Engineering (IFIN-HH) Bucharest 077125, Romania c

Institute of Theoretical Physics, Faculty of Physics, University of Warsaw ul. Pasteura 5, 02-093 Warsaw, Poland Abstract

Spontaneous breaking of quantum scale invariance may provide a solution to the hierarchy and cosmological constant problems. In a scale-invariant regularization, we compute the two-loop potential of a higgs-like scalar φ in theories in which scale symmetry is broken only spontaneously by the dilaton (σ). Its vev hσi generates the DR subtraction scale (µ ∼ hσi), which avoids the explicit scale symmetry breaking by traditional regularizations (where µ=fixed scale). The two-loop potential contains effective operators of non-polynomial nature as well as new corrections, beyond those obtained with explicit breaking (µ=fixed scale). These operators have the form: φ6 /σ 2 , φ8 /σ 4 , etc, which generate an infinite series of higher dimensional polynomial operators upon expansion about hσi ≫ hφi, where such hierarchy is arranged by one initial, classical tuning. These operators emerge at the quantum level from evanescent interactions (∝ ǫ) between σ and φ that vanish in d = 4 but are demanded by classical scale invariance in d = 4 − 2ǫ. The Callan-Symanzik equation of the two-loop potential is respected and the two-loop beta functions of the couplings differ from those of the same theory regularized with µ =fixed scale. Therefore the running of the couplings enables one to distinguish between spontaneous and explicit scale symmetry breaking.

∗

E-mail: [email protected], [email protected], [email protected]

1

Introduction

Theories with scale symmetry [1] may provide a solution to the hierarchy and cosmological constant problems. But scale symmetry is not a symmetry of the real world, therefore it must be broken. In this work we discuss theories with scale invariance at the classical and quantum level that is broken only spontaneously. This is important since in a classical scale invariant theory, quantum calculations usually break this symmetry explicitly due to the presence of the subtraction (renormalization) scale (µ). This scale is introduced to regularize the loop integrals, regardless of the regularization method: dimensional regularization (DR), Pauli-Villars, etc, and its simple presence breaks explicitly this symmetry. It is known however how to avoid this problem by using a subtraction scale that is generated spontaneously, as the vacuum expectation value (vev) of a scalar field σ [2]. This field is the Goldstone mode of scale symmetry (dilaton) and then µ = zhσi, where z is a dimensionless parameter. But before (spontaneous) scale symmetry breaking, with a fielddependent subtraction function µ(σ) = zσ, there is no scale in the theory. One can use this idea to compute quantum corrections to the scalar potential of a theory with a higgs-like scalar φ and dilaton σ and obtain a scale invariant result at one-loop [3, 4, 5, 6, 7] with a flat direction and spontaneous scale symmetry breaking. Although the result is scale invariant at the quantum level, the couplings still run with the momentum scale [5, 6, 8]1 . To illustrate some of these ideas, consider a scale invariant theory in d = 4 L=

1 1 ∂µ φ∂ µ φ + ∂µ σ∂ µ σ − V (φ, σ) 2 2

(1)

where φ is a higgs-like scalar and σ is a dilaton. In such a theory V has a form V (φ, σ) = σ 4 W (φ/σ)

(2)

In this paper we assume that we have spontaneous breaking of this symmetry, hence hσi = 6 0. We do not detail how σ acquires a vev (expected to be large hσi ∼ MPlanck ) but search for solutions with hσi = 6 0. Then the two minimum conditions ∂V /∂φ = ∂V /∂σ = 0 become W ′ (x0 ) = W (x0 ) = 0,

x0 ≡

hφi ; hσi

hσi, hφi 6= 0.

(3)

At a given order n in perturbation theory, one condition, say W ′ (x0 ) = 0, fixes the ratio x0 ≡ hφi/hσi in terms of the (dimensionless) couplings of the theory. The second condition, W (x0 ) = 0, leads to vanishing vacuum energy V (hφi, hσi) = 0 and fixes a relation among the couplings, corrected to that order (n) in perturbation theory from its version in the lower perturbation order (n − 1). If these two equations have a solution x0 , then the system has a flat direction (Goldstone) in the plane (φ, σ) with φ/σ = x0 . Then a massless state exists 1 After spontaneous breaking of scale symmetry hσi = 6 0, the subtraction scale µ(hσi) and all other masses/vev’s of the theory are generated, proportional to hσi.

1

(dilaton) at this order. This is true provided that quantum corrections do not break explicitly the scale symmetry (otherwise, eq.(2) is not valid due to the presence of the “usual” DR scale µ). With a scale invariant regularization, it is possible to keep these properties (V = 0, a flat direction, etc) and study spontaneously broken quantum scale invariance. Why is this interesting? One reason is that this answers the question of Bardeen [9] on the mass hierarchy. The Standard Model (SM) with a vanishing classical higgs mass term is scale invariant and there is no mass hierarchy (ignoring gravity, as here2 ). If quantum calculations preserve this symmetry, via a scale invariant regularization, one can avoid a hierarchy problem and the fine-tuning of the higgs self-coupling and keep it light relative to the high scale (physical mass of a new state) generated by hσi 6= 0. One can arrange that x0 = hφi/hσi ≪ 1 by a single classical tuning of the (ratio of the) couplings of the theory [14]. The hierarchy m2higgs ∼ hφi2 ≪ hσi2 is maintained at one-loop [3, 4, 5, 6, 7, 14, 15] and probably beyond it, due to the spontaneous-only scale symmetry breaking. The only difference from the usual SM is the presence of a massless dilaton in addition to the SM spectrum. Also, the solution x0 is related to the (minimum) condition V = 0. This suggests that in spontaneously broken quantum scale invariant theories any fine tuning is related to vacuum energy tuning at the same order of perturbation. With this motivation, in this paper we extend the above results. We consider a classically scale invariant theory of φ and σ and compute at two-loop the scalar potential and the running of the couplings, in a scale invariant regularization. We find that starting from two loops, the running of the couplings differs from that in the same theory of φ, σ regularised with µ =constant. We show that effective non-polynomial operators like φ6 /σ 2 , φ8 /σ 4 , are generated as two-loop counterterms. If expanded about the ground state, these operators generate an infinite series of polynomial terms, showing the non-renormalizability of the theory. The Callan-Symanzik equation of the potential is verified at two loops. The results are useful for phenomenology, e.g. to study a scale invariant version of the SM (+dilaton).

2

One-loop potential

We first review the one-loop potential [6, 7]. Consider the classical potential3 V =

λφ 4 λm 2 2 λσ 4 φ + φ σ + σ . 4! 4 4!

(4)

Spontaneous scale symmetry breaking hσi 6= 0 demands two conditions (eq.(3)) be met: 9λ2m = λφ λσ + loops, (λm < 0),

and

x20 ≡

hφi2 3λm =− + loops. hσi2 λφ

(5)

A massless (Goldstone) state exists corresponding to a flat direction φ = x0 σ with Vmin = 0. With φ being higgs-like, scale symmetry breaking implies electroweak symmetry breaking. 2 3

For related applications that include gravity, see for example [10, 11, 12, 13]. In principle one can also include higher dimensional terms like φ6 /σ 2 , φ8 /σ 4 , etc, (hσi = 6 0), see later.

2

To compute quantum corrections in d = 4 − 2ǫ, the scalar potential is modified to V˜ = µ2ǫ V to ensure dimensionless quartic couplings, with µ the “usual” DR subtraction scale. General principles4 suggest that the subtraction function µ(σ) depend on the dilaton only [6] and generate the subtraction scale µ(hσi) after spontaneous scale symmetry breaking; µ(σ) is then identified on dimensional grounds (using [µ] = 1, [σ] = (d − 2)/2). Then the scale invariant potential in d = 4 − 2ǫ and µ(σ) become V˜ (φ, σ) = µ(σ)2ǫ V (φ, σ),

µ(σ) = z σ 1/(1−ǫ) ,

(6)

where z is an arbitrary dimensionless parameter5 . The one-loop result is V1

i = V˜ − 2

Z

  dd p Tr ln p2 − V˜αβ + iε d (2π)

(7)

Here V˜ij = ∂ 2 V˜ /∂si ∂sj , (i, j = 1, 2), s1 = φ, s2 = σ and similar for Vij = ∂ 2 V /∂si ∂sj .   Also V˜ij = µ2ǫ Vij + 2ǫ µ−2 Nij + O(ǫ2 ), where Nij ≡ µ

n ∂µ ∂V ∂µ ∂V o n ∂ 2 µ ∂µ ∂µ o + − + µ V, ∂si ∂sj ∂sj ∂si ∂si ∂sj ∂si ∂sj

i, j = 1, 2.

(8)

Then n V1 = µ(σ)2ǫ V −

 Ms2 (φ, σ)  4 (Vij Nji ) io 1 h X 4 1 + − ln M s 64π 2 ǫ c0 µ2 (σ) µ2 (σ)

(9)

s=φ,σ

with an implicit sum over i, j and with c0 = 4πe3/2−γE . The one-loop Lagrangian is 1 1 L1 = (∂µ φ)2 + (∂µ σ)2 − V1 . 2 2

(10)

Above, Ms2 denotes the field-dependent eigenvalues of the matrix Vij . The poles in L1 are cancelled by adding the counterterm Lagrangian δL1 found using the expression of the Ms2 :

δL1 ≡ −δV1 = −µ(σ)2ǫ

n1

with

4 5

 1 1 (Zλφ − 1)λφ φ4 + (Zλm − 1)λm φ2 σ 2 + (Zλσ − 1)λσ σ 4 (11) 4! 4 4!

Zλφ

= 1+

3 (λφ + λ2m /λφ ), 2κ ǫ

Zλm

= 1+

1 (λφ + λσ + 4λm ), 2κ ǫ

They demand quantum interactions between φ and σ vanish in their classically decoupling limit λm = 0. The parameter z plays a special role in the Callan Symanzik equation, see later.

3

= 1+

Zλσ

3 (λσ + λ2m /λσ ), 2κ ǫ

κ = (4π)2 .

(12)

Zλ ’s are identical to their counterparts computed in the same theory regularized with µ=constant (when scale symmetry is broken explicitly). The one-loop potential becomes U1 = V + V (1) + V (1,n) , V (1) ≡ V (1,n) ≡

(13)

h M 2 (φ, σ) 3 i X 1 4 , − M (φ, σ) ln s 2 s 2 64π µ (σ) 2

(14)

s=φ,σ

i 1 h φ6 λφ λm 2 −(16λφ λm + 18λ2m −λφ λσ )φ4 −(48λm +25λσ )λm φ2 σ 2 −7λ2σ σ 4 .(15) 48κ σ

The potential simplifies further if we use the tree-level relation (5) among λs (s = φ, m, σ) that ensures the spontaneous scale symmetry breaking. U1 is scale symmetric and a flat (1,n) direction exists also at the quantum level. V1 is a new, finite one-loop correction, independent of the parameter z; it contains a non-polynomial term φ6 /σ 2 that can be Taylorexpanded about hφi, hσi = 6 0. V (1,n) → 0 in the classical decoupling limit λm → 0. The Coleman-Weinberg term is also present, with µ → µ(σ) and thus depends on z. This dependence replaces the “traditional” dependence of V (1) on the subtraction scale in theories regularized with µ =constant. But physics should be independent of this parameter, which means that U1 must respect the Callan-Symanzik equation: dU1 /d ln z = 0 [8]. To check the Callan-Symanzik equation, we need the beta functions of the couplings which run with the momentum, even in scale invariant theories [5, 8]. These are computed from the condition d(µ(σ)2ǫ λj Zλj )/d ln z = 0 (j: fixed), since the bare coupling is independent of z. The result is identical to that in a theory regularized with µ=constant: βλφ

(1)

≡

dλφ 3 = (λ2φ + λ2m ), d ln z κ

(16)

βλm

(1)

≡

1 dλm = (λφ + 4λm + λσ )λm d ln z κ

(17)

(1)

≡

3 dλσ = (λ2m + λ2σ ). d ln z κ

(18)

βλσ

The Callan Symanzik equation at one-loop is dU1 (λj , z)  (1) ∂ ∂  U1 (λj , z) = O(λ3j ), = βλj +z d ln z ∂λj ∂z

(sum over j = φ, m, σ).

(19)

Eq.(19) is easily verified with the above results for the beta functions. The one-loop U1 can be used for phenomenology of a scale invariant version of the SM extended by the dilaton [6].

4

3

Two-loop potential

3.1

New poles in the two-loop potential

To compute the two-loop potential we use the background field expansion method about φ, σ. We Taylor-expand V˜ about these values 1 1 1 V˜ (φ + δφ, σ + δσ) = V (φ, σ) + V˜j sj + V˜jk sj sk + V˜ijk si sj sk + V˜ijkl si sj sk sl + · · · (20) 2 3! 4! where the subscripts i, j, k, l of V˜ij... denote derivatives of V˜ wrt fields of the set {φ, σ}j ; with i, j, k, l = 1, 2. Also s1 = δφ, s2 = δσ are field fluctuations. Notice that there are new, evanescent interactions (∝ ǫ) in vertices V˜ijk··· generated by eq.(6) that impact on the loop corrections. The two-loop diagrams are presented below. Let us first denote: V2 = V2a + V2b + V2c .

(21)

Then V2a

Z

dd p (2π)d

Z

dd q ˜ −1 ˜ q−1 )jm (D ˜ −1 )kn (Dp )il (D p+q (2π)d

=

i 12

=

µ(σ)2ǫ 1 h 4 3 φ (λφ + λφ λ2m + 2λ3m ) + σ 4 (2λ3m + λ2m λσ + λ3σ ) ǫ2 16κ2

i = V˜ijk V˜lmn 12

i + φ2 σ 2 (λ2φ λm + 6λφ λ2m + 10λ3m + 6λ2m λσ + λm λ2σ ) + O(1/ǫ). (22)

Also V2b =

=

i 8

=

i˜ h Vijkl 8

Z

dd p ˜ −1 i h (D )ij (2π)d p

Z

dd q ˜ −1 i (D )kl (2π)d q

µ(σ)2ǫ 1 h 4 3 φ (λφ + 2λφ λ2m + λ2m λσ ) + σ 4 (λφ λ2m + 2λ2m λσ + λ3σ ) ǫ2 32κ2 i + 2λm φ2 σ 2 (λ2φ + 9λ2m + λφ λσ + λ2σ ) + O(1/ǫ),

(23)

and finally

V2c

=

i 2

i = (δV1 )ij 2

Z

dd p ˜ −1 (Dp )ij (2π)d

µ(σ)2ǫ (−1) h 4 φ (3λ3φ + 4λφ λ2m + 4λ3m + λ2m λσ ) + σ 4 (λφ λ2m + 4λ3m + 4λ2m λσ + 3λ3σ ) ǫ2 16κ2 i + φ2 σ 2 (4λ2φ λm + 12λφ λ2m + 38λ3m + 2λφ λm λσ + 12λ2m λσ + 4λm λ2σ ) + O(1/ǫ). (24)

=

5

These diagrams are computed using [17], see also [18]. The propagators are given by the ˜ p )ij = p2 δij − V˜ij . To simplify the calculation they can be re-written inverse of the matrix (D ˜ −1 )ij = a as (D ˜ij /(p2 − V˜p ) + ˜bij /(p2 − V˜m ), with appropriate coefficients a ˜ij and ˜bij and where V˜p , V˜m (V˜p > V˜m ) denote the field-dependent masses, eigenvalues of the matrix V˜ij , i, j = φ, σ. Note that V˜ij , V˜p , V˜m , a ˜ij , ˜bij and also V˜ijkl , V˜ijk contain positive powers of ǫ; this is relevant for the above calculation, since they contribute to the finite and 1/ǫ parts of the potential. Their form is detailed in Appendix B and C. One notices that the poles 1/ǫ2 in V2a,b,c are identical to those in the theory regularized with µ =constant. This is expected for this leading singularity, but this is not true for their sub-leading one (1/ǫ) or for their finite part (see later). The long expressions O(1/ǫ) and O(ǫ0 ) of each diagram V2a,b,c are not shown here. The sum of these diagrams gives V2 =

+

µ(σ)2ǫ (−1) h 4 φ (3λ3φ +4λφ λ2m + 4λ3m + λ2m λσ ) +σ 4 (3λ3σ + λφ λ2m +4λ3m + 4λ2m λσ ) ǫ2 32κ2 i + φ2 σ 2 (4λ2φ λm + 12λφ λ2m + 38λ3m + 2λφ λm λσ + 12λ2m λσ + 4λm λ2σ )

µ(σ)2ǫ 1 h 4 3 (25) φ (λφ + λφ λ2m + 2λ3m ) + σ 4 (2λ3m + λ2m λσ + λ3σ ) ǫ 16κ2 i 1/ǫ +φ2 σ 2 (λ2φ λm + 6λφ λ2m + 10λ3m + 6λ2m λσ + λm λ2σ ) + V2 + V (2) + V (2,n) .

Here V (2) and V (2,n) are O(ǫ0 ) i.e. finite quantum corrections presented in Appendix B. 1/ǫ V2 = O(1/ǫ) is a new term that contains 1/ǫ poles not present in the theory regularized with µ=constant; its origin is due to evanescent interactions (∝ ǫ), which “meet” 1/ǫ2 poles, thus giving 1/ǫ terms. One finds 1/ǫ

V2

 7 4 1 2 7 2 1 µ(σ)2ǫ h 4  20 2 2 3 2 φ λ + λ − λ λ λ − 2λ − λ λ λ λ + λ λ + λ λ m σ m σ σ φ φ φ m m σ 16κ2 ǫ 3 φ 6 2 φ 3 12 m 4    41 1 43 1 7  + φ2 σ 2 8λφ λ2m + λ3m + λφ λm λσ + λ2m λσ + λm λ2σ + σ 4 4λ3m + λ2m λσ + λ3σ 2 3 2 3 4  1 8 i 7 1 φ6  7 2 2 2 φ − λ + λ λ λ − λ λ λ λ λ − (26) + m m σ φ φ φ m m 4 . σ2 6 φ 3 6 4 σ

=

In addition to usual counterterms (φ4 , etc), notice from eq.(26) the need for non-polynomial counterterms φ6 /σ 2 and φ8 /σ 4 (see also φ6 /σ 2 in eq.(13)). The above two-loop results contribute to the Lagrangian (below ρφ , ρσ are wavefunction coefficients defined later) L2 =

  1  ρφ 1  ρσ + finite (∂µ φ)2 + + finite (∂µ σ)2 − V2 . 2 ǫ 2 ǫ

(27)

A counterterm δL2 cancels the poles in the sum L1 + L2 of eqs.(10), (27) up to two-loops 6

δL2 =

n λφ 1 1 (Zφ − 1)(∂µ φ)2 + (Zσ − 1)(∂µ σ)2 − µ(σ)2ǫ (Zλφ − 1) φ4 + 2 2 4!

+ (Zλm − 1)

λm 2 2 λσ λ6 φ6 λ8 φ8 o φ σ + (Zλσ − 1) σ 4 +(Zλ6 − 1) +(Z − 1) , (28) λ 8 4 4! 6 σ2 8 σ4

where Zλφ Zλm Zλσ Zλ6

1  δφ + ν1φ δ2φ  δ0φ + 2 1 + 2 , κǫ κ ǫ ǫ  δm 1 δm + ν1m δ2m  = 1+ 0 + 2 1 + 2 , κǫ κ ǫ ǫ 1  δσ + ν1σ δσ  δσ + 22 , = 1+ 0 + 2 1 κǫ κ ǫ ǫ = 1+

= 1+

1 ν16 ; κ2 ǫ

Zλ8 = 1 +

1 ν18 , κ2 ǫ

(29)

where one-loop δ0φ , δ0m , δ0σ can be read from eq.(12) while the two-loop coefficients δks , k = 1, 2, s = φ, m, σ, are shown in Appendix A. They are obtained by comparing δL2 against L2 , using V2 of eq.(25). The coefficients δks are those of the theory regularized with µ =constant. However, there is an extra contribution from coefficients ν1s , s = φ, m, σ, 6, 8 1/ǫ (see Appendix A), that is generated by the new poles 1/ǫ of V2 . This new contribution brings a correction to the two-loop beta functions of our theory, see later. One can also show that the two-loop-corrected wavefunction coefficients have expressions similar to those in the theory regularised with µ =constant:

Zφ = 1 +

ρφ , κ2 ǫ

ρφ = −

Zσ = 1 +

ρσ , κ2 ǫ

ρσ = −

1 2 (λ + 3λ2m ), 24 φ 1 (λ2 + 3λ2m ), 24 σ

(30)

One often uses the notation γφ = −2ρφ /κ2 and γσ = −2ρσ /κ2 for the anomalous dimensions.

3.2

Two-loop beta functions

With the above information, one obtains the two-loop beta functions. To this purpose, one uses that the “bare” couplings λB j below are independent of the parameter z: λB = µ(σ)2ǫ λφ Zλφ Zφ−2 , φ −1 −1 2ǫ λB m = µ(σ) λm Zλm Zφ Zσ ,

λB = µ(σ)2ǫ λσ Zλσ Zσ−2 , σ λB = µ(σ)2ǫ λ6 Zλ6 Zφ Zσ−3 . 6 7

(31)

6 We thus demand that (d/d ln z)λB k = 0, k = φ, m, σ, 6, 8 . Taking the logarithm of the first expression in (31) and then the derivative with respect to ln z, one obtains

2ǫ +

βλφ + λφ

X

βλj

j=φ,m,σ

i h d ln Zλφ Zφ−2 = 0 d ln z

(32)

and similar expressions for the other couplings. Using the form of Z ′ s, one finds βλφ = −2ǫλφ + 2λφ

X

λj

j=φ,m,σ

d  δ0φ δ1φ + ν1φ 2 ρφ  + − 2 . dλj κ κ2 κ

(33)

One easily obtains similar relations for βλm and βλσ (for βλσ just replace the sub-/superscript φ → σ). The difference in these beta functions from those in the same theory but regularized with µ =constant is the presence of a new contribution: ν1φ (ν1m , ν1σ , respectively), that we identified in eqs.(29). Eq.(33) is solved with particular attention to the ǫ-dependent terms, to find at two-loop: βλφ

=

1 17 3 2 (2,n) (λφ + λ2m ) − 2 ( λ3φ + 5λφ λ2m + 12λ3m ) + βλφ , κ κ 3

βλm

=

1 λm (2,n) (λφ + 4λm + λσ )λm − 2 (5λ2φ + 36λφ λm + 54λ2m + 36λm λσ + 5λ2σ ) + βλm , κ 6κ

βλσ

=

1 17 3 2 (2,n) (λ + λ2σ ) − 2 (12λ3m + 5λ2m λσ + λ3σ ) + βλσ . κ m κ 3 (2,n)

The “new” terms βλ

on the rhs are

 1  2 2 2 2 (−80λ + 6λ ) , λ (24λ − 7λ ) + λ (−14λ + 16λ λ − 3λ ) + λ m σ m σ m σ φ m m σ φ 2κ2

(2,n)

=

(2,n)

= −

λm (48λφ λm + 6λφ λσ + 123λ2m + 86λm λσ + 3λ2σ ), 6κ2

(2,n)

= −

1 (48λ3m + 4λ2m λσ + 21λ3σ ), 2κ2

(2,n)

=

1 λφ λm (7λφ − 14λm + λσ ), 4κ2

(2,n)

=

1 λφ λ2m . 2κ2

βλφ

βλm βλσ βλ6 βλ8

(34)

(35)

(2,n)

Here βλ that appears for each λ at two-loop is the mentioned correction, that is missed if this theory is regularized with µ =constant, when one breaks explicitly the scale symmetry. Notice that λ6,8 also run in this order in the scale invariant theory. 6 We also include the effect of wavefunction renormalization of the subtraction function which demands 1/2 replacing: µ(σ) = z σ 1/(1−ǫ) → z (Zσ σ)1/(1−ǫ) ; however, this brings no correction in this order.

8

We conclude that from the two-loop running of the couplings, encoded by the beta functions, one can distinguish between the theory with (spontaneously broken) scale symmetry at quantum level and that in which this symmetry is broken explicitly by quantum corrections (with µ =constant). There is a simple way to understand this difference: the theory regularized with µ =constant, and two fields φ, σ is renormalizable while our model, scale invariant at quantum level, is non-renormalizable. This is due to the scale-invariant nonpolynomial terms of type φ6 /σ 2 , φ8 /σ 4 generated at one- and two-loop level7 . This justifies the different beta functions in the two approaches starting from the two-loop level. This is an interesting result of the paper.

3.3

Two-loop potential after renormalization

Finally, we present the two-loop potential U after renormalization. It has the form (1) (1,n) (2) (2,n) U =V | + V {z+ V } +V + V

(36)

=U1

where U1 is the one-loop result of (13). V (2) is a two-loop correction identical to that obtained in the theory regularized with µ =constant (up to replacing µ → z σ), while V (2,n) are new two-loop terms that involve derivatives of µ(σ) wrt σ (similar to one-loop V (1,n) )8 The long expressions of V (2) , V (2,n) are given in Appendix B, eq.(B-5). U contains new, non-polynomial effective operators, such as φ6 /σ 2 and φ8 /σ 4 , etc: U=

5λ3φ φ6 7λ3φ φ8 + + ··· 576 κ2 σ 4 24 κ2 σ 2

(37)

All non-polynomial terms present in the potential can be expanded about the ground state φ = hφi + δφ,

σ = hσi + δσ

(38)

where δφ and δσ represent fluctuations about the ground state. Then each non-polynomial operator becomes an infinite series expansion about the point hφi/hσi. For example   2  φ6 δφ2 2δσ 3δσ 2 2δφ 4 hφi + + = (hφi + δφ) + · · · 1 − + · · · 1 + σ2 hσi2 hφi hφi2 hσi hσi2

(39)

and similarly for the operator φ8 /σ 4 in U , etc. Although we did not present the ground state of the one-loop potential, this is known to satisfy the relation [6]: hφi2 /hσi2 = −3λm /λσ (1+ loop-corrections) [6]. Using this information in eq.(39) and (37), one sees that in the classical decoupling limit λm → 0, the non-polynomial operators of (37) do vanish. 7 8

This non-renormalizability argument is different from that in [4] which does not apply here, see [6]. See [19] for further discussion on the Goldstone modes contributions to the potential.

9

It is important to stress that only operators of the form φ2n+4 /σ 2n , n ≥ 1 were generated in the two-loop potential, but no operator like σ 2n+4 /φ2n , n ≥ 1 is present. This is due to the way the subtraction function enters in the loop corrections, via derivatives wrt σ of µ(σ)ǫ which are suppressed by positive powers of µ(σ). This means that all higher dimensional operators are ultimately suppressed by (large) hσi and not proportional to it. This is welcome for the hierarchy problem, since such terms could otherwise lead to corrections to the higgs mass of the type λ3φ hσi2 requiring tuning the higgs self-coupling λφ , and thus re-introducing the hierarchy problem. This problem is avoided at least at one-loop [3, 6].

4

Two-loop Callan-Symanzik for the potential

A good check of our two-loop scale-invariant potential is the Callan-Symanzik equation, in its version for scale invariant theories [8]. This equation states the independence of the two-loop potential of the subtraction (dimensionless) parameter z; this parameter fixes the subtraction scale to zhσi, after spontaneous scale symmetry breaking. The equation is d U (λ, z)  ∂ ∂  ∂ ∂ U (λ, z) = 0 , = z + βλj − σγσ − φγφ d ln z ∂z ∂λj ∂φ ∂σ

(40)

where the j-summation runs over λj = λφ , λm , λσ , λ6 , λ8 . Eq.(40) can be re-written as a set of equations at a given order of λ’s (or number of loops). To help one trace the difference between our scale-invariant result and that of the same theory but with µ =constant, below we use for U the decomposition given in eq.(36) while for the beta functions we write βλj

(1)

(2)

(2,n)

= βλj + βλj + βλj (1)

.

(41) (2)

The terms in the beta function correspond to 1-loop (βλ ), 2-loop-only (βλ ) and 2-loop-new (2,n) parts (βλ ). Then, with the two-loop anomalous dimensions γφ , γσ defined after eq.(30), a careful analysis shows that eq.(40) splits into ∂ V (1) (1) ∂V + βλj =0 ∂ ln z ∂λj

(42)

∂ V (1,n) =0 ∂ ln z   (1) ∂ ∂ ∂ V (2) (2) ∂ (1) ∂V + βλj − γφ φ − γσ σ =0 V + βλj ∂ ln z ∂λj ∂φ ∂σ ∂λj (1,n) ∂ V (2,n) (2,n) ∂V (1) ∂V + βλj + βλj =0. ∂ ln z ∂λj ∂λj

(43) (44)

(45)

where V includes the new terms (λ6 /6) φ6 /σ 2 + (λ8 /8) φ8 /σ 4 . We checked that these equa10

tions are respected. Eqs.(42), (44) express the usual Callan-Symanzik equation (of the theory with µ =constant), whereas (43) and (45) constitute a new part, which is nonzero only when µ = µ(σ). Eq.(43) is obvious and hardly revealing. But checking eq.(45) is more difficult. For this one also needs to take account of the “new” corrections to two-loop beta functions of λ6,8 , see eq.(35) and also the z-dependent part of V (2,n) which we write below V (2,n) = −

ih
1 h lnV + lnV − 144λ2φ λm − 111λφ λ2m − 168λ3m + 9λ2φ λσ − 40λ2m λσ φ4 p m 2 192 κ
192λφ λ2m + 705λ3m + 37λφ λm λσ + 368λ2m λσ + 106λm λ2σ φ2 σ 2


φ6
48λ3m + 46λ2m λσ + 63λ3σ σ 4 + 18λ2φ λm − 24λφ λ2m + 3λ3m + 3λφ λm λσ 2 σ A φ8 i + 3λφ λ2m 4 + z-independent terms, where lnA ≡ ln − 1. (46) 2 σ (zσ) 4πe−γE

−

where γE = 0.5772..... Here Vp and Vm are field dependent eigenvalues of the matrix of second derivatives Vij wrt i, j = φ, σ of the tree level potential. Given this, the CallanSymanzik equation of the potential is verified at the two-loop level.

5

Conclusions

Quantum scale invariance with spontaneous breaking may provide a solution to the cosmological constant and the hierarchy problem. The “traditional” method for loop calculations breaks explicitly classical scale symmetry of a theory due to the regularization which introduces a subtraction scale (DR scale, cut-off, Pauli-Villars scale). However, it is known how to perform quantum calculations in a manifestly scale invariant way: the subtraction scale is replaced by a subtraction function of the field(s) (dilaton σ) which when acquiring a vev spontaneously, generates this scale µ(hσi) = zhσi. The Goldstone mode of this symmetry is the dilaton field which remains a flat direction of the quantum scale-invariant potential. Starting with a classically scale-invariant action, we computed the two-loop scalar potential of φ (higgs-like) and σ in a scale invariant regularization. The one- and two-loop potential are scale invariant and contain new terms beyond the usual corrections obtained for µ =constant (Coleman-Weinberg, etc), due to field derivatives of µ(σ). They also contain interesting effective non-polynomial operators φ6 /σ 2 and φ8 /σ 4 , etc, allowed by scale symmetry, showing that such theories are non-renormalizable. These operators can be expanded about the non-zero hφi and hσi, to obtain an infinite series of effective polynomial ones, suppressed by hσi ≫ hφi (such hierarchy can be enforced by one initial, classical tuning of the couplings). The non-polynomial operators emerge from evanescent interactions (∝ ǫ) between φ and σ that vanish in d = 4 but are demanded by scale invariance in d = 4 − 2ǫ. Previous works also showed that the higgs mass is stable against quantum corrections at one-loop m2φ ≪ hσi2 , which may remain true beyond it if only spontaneous scale symmetry breaking is present. 11

We checked the consistency of the two-loop scale invariant potential by showing that it satisfies the Callan-Symanzik equation in its scale-invariant formulation. To this purpose we computed the two-loop beta functions of the couplings of the theory. While one-loop beta functions are exactly those of the same theory of φ, σ regularized with µ =constant, the two-loop beta functions differ from those of the theory regularized with explicit breaking of scale symmetry (µ =constant). In conclusion, the running of the couplings enables one to distinguish between spontaneous and explicit breaking of quantum scale symmetry of the action. ——————————–

Appendix: A

Coefficients of the counterterms

Assuming λn = 0, n > 6 at tree-level, the coefficients of eqs.(29), (33) are: δ1φ = −

3 2



λ2φ + λ2m + 2

λ3m λφ



(A-1)


1 2 λφ + 6λφ λm + 10λ2m + 6λm λσ + λ2σ (A-2) 4  2  λm 3 2 2 + λm + λσ 2 (A-3) = − 2 λσ   1 λ2m 2 2 (24λm − 7λσ ) − (14λm − 16λm λσ + 3λσ ) − λφ (80λm − 6λσ ) (A-4) = 8 λφ

δ1m = − δ1σ ν1φ

ν1m = − ν1σ = − ν16 = ν18 = δ2φ = δ2m = δ2σ =

 1  123λ2m + 86λm λσ + 3λ2σ + 6λφ (8λm + λσ ) 24
1 48λ3m + 4λ2m λσ + 21λ3σ 8λσ

1 λφ λm (7λφ − 14λm + λσ ) 16 λ6

(A-5) (A-6) (A-7)

1 λφ λ2m 8 λ8   3 λ2m 2 2 (4λm + λσ ) 3λφ + 4λm + 4 λφ
1 2λ2φ + 6λφ λm + λφ λσ + 19λ2m + 6λm λσ + 2λ2σ 4   3 λ2m 2 2 (λφ + 4λm ) + 4λm + 3λσ . 4 λσ 12

(A-8) (A-9) (A-10) (A-11)

B

The finite part of the two-loop potential

We provide here the finite part of the two-loop potential, V (2) + V (2,n) of eq.(25), (36). This is rather long, we thus use a simplified notation. The propagators are found from: e p )ij = p2 δij − Veij . To simplify the calculation it helps to write them as (D e −1 )ij = (D

a ˜ij

p2 − Vep

Vep − Ve22 , a ˜11 = ˜b22 = Vep − Vem

+

˜bij

p2 − Vem

˜bij = δij − a ˜ij

,

a ˜22 = ˜b11 = 1 − a ˜11 =

Vep − Ve11 , Vep − Vem

a ˜12 = a ˜21 =

Ve12 , (B-1) Vep − Vem

where Vep , Vem are the field-dependent eigenvalues of matrix Veij = ∂ 2 Ve /∂si ∂sj , i, j = 1, 2; s1 = φ, s2 = σ, and Ve = µ(σ)2ǫ V where V is the tree level potential in d = 4. We introduce the following coefficients (without e) of the Taylor expansions in ǫ (see Appendix C for their values in terms of the couplings and fields): a ˜ij

= aij + ǫ a1ij + ǫ2 a2ij + O(ǫ3 ),

  Veijk... = µ(σ)2ǫ vijk... + ǫ uijk... + ǫ2 wijk... + O(ǫ3 ) , ∂ 4 Ve ∂4V , vijk... = , ∂si ∂sj ∂sk · · · ∂si ∂sj ∂sk · · ·   = µ(σ)2ǫ Vp 1 + c1p ǫ + c2p ǫ2 + O(ǫ3 )

Veijk... = Vep

b1ij = −a1ij ,

bij = δij − aij ,

b2ij = −a2ij

where:

i, j, k, . . . = 1, 2;

  Vem = µ(σ)2ǫ Vm 1 + c1m ǫ + c2m ǫ2 + O(ǫ3 ) .

s1 = φ, s2 = σ,

(B-2)

Here Vp and Vm are the field-dependent eigenvalues of the matrix Vij of the-tree level V : h   i 2 1/2 Vp = 1/2 V11 + V22 + (V11 − V22 )2 + 4V12

(B-3)

with Vm of similar expression but with − in front of the square root. Vp and Vm should not be confused with derivatives Vi of the potential. We also use the notation lnA = ln

A − 1, t(zσ)2

t = 4πe−γE .

(B-4)

Then V (2) and V (2,n) of eq.(25), (36) are shown below. V (2) is that of the theory regularized with µ =constant, while V (2,n) is a new correction. They are sums of the diagrams of eq.(22) c b a + V2,old + V2,old V (2) = V2,old a b c V (2,n) = V2,n + V2,n + V2,n

(B-5)

where a, b, c, label the sunset (a), snowman (b), counterterm (c) diagrams, respectively. 13

Then, in terms of the above coefficients, one finds for the sunset diagram (a): i    1 nh 1 u u v w + Vp ail + Vm bil ajm + bjm akn + bkn ij ijk lmn lmn 2 4κ 2 h  
  + vijk ulmn Vp ail (1−2lnVm +c1p ) + a1il + Vm bil (1−2lnVm +c1m ) + b1il ajm +bjm

a V2,n =

h1      i + 2 Vp ail + Vm bil a1jm + b1jm akn + bkn + vijk vlmn Vp ail (c2p − c1p ) + a1il c1p 2 
  − 2lnVp (ail c1p + a1il ) + Vm bil (c2m − c1m ) + b1il c1m − 2lnVm (bil c1m + b1il ) ajm + bjm +

      1
i ajm + b1jm akn + bkn Vp ail c1p − 2lnVp + Vm bil c1m − 2lnVm

h     + vijk vlmn Vp bil 2a1jm a1kn + a2jm bkn + a1jm bkn + 2a1jm b1kn + ail 3a1jm a1kn + 4a2jn bkn    + 2bjm b1kn + 4a1jm (bkn + b1kn ) + 2bjm b2kn + ail ajm 3(a1kn + a2kn ) + 2(b1kn + b2kn )

    + Vm ail 2b1jm b1kn + b2jm akn + b1jm akn + 2b1jm a1kn + bil 3b1jm b1kn + 4b2jn akn + 2ajm a1kn

and

 i o   + 4b1jm (akn + a1kn ) + 2ajm a2kn + bil bjm 3(b1kn + b2kn ) + 2(a1kn + a2kn )

a = V2,old

(B-6)

n   1 v v Vp ail lnVp + Vm bil lnVm ajm akn lnVp + bjm bkn lnVm ijk lmn 2 4κ

 2    1 2 2 2 + 2 vp ail ln Vp + Vm bil ln Vm ajm bkn + Vp bil − Vm ail ln Vp − ln Vm ajm bkn 2     − Vp ail lnVp + Vm bil lnVm ajm + bjm akn + bkn

 h 3 π 2 i  + Vp ail + Vm bil ajm + bjm akn + bkn 2 12 h

i 1 1 − Vp 2ail Φp,m − bil Φm,p + Vm 2bil Φm,p − ail Φp,m ajm bkn 2 2 o  1 Vp ail ajm akn + Vm bil bjm bkn C , − 3 +

with [17]

Φp,m



q  h    y ypm−1 − 4 Li2 1−η2pm + 2 ln2 pm = q
√ ypm  4 1−y , y arcsin Cl pm 2 pm

ypm = Vm /Vp , Li2 (ξ) = −

Z

0

1

dt

− ln2 4ypm +

π2 3

i ,

ypm > 1

(B-8) , 1 > ypm > 0

√ C = −2 3Cl2 (π/3) ∼ = 3.5 Z θ θ dθ ln 2 sin . Cl2 (θ) = − 2 0

ηpm = (1 − 1/ypm )1/2 ,

ln(1 − ξt) , t

1−ηpm 2

(B-7)

14

(B-9) (B-10)

Further, for the snowman diagram (b): h 1 n w (a V + b V ) (a V + b V ) − 2 u ij p ij m ijkl kl p kl m ijkl (aij Vp + bij Vm ) 8 κ2  i  × akl Vp lnVp + bkl Vm lnVm − (akl cp1 + a1kl ) Vp + (bkl cp1 + b1kl ) Vm

b V2,n =

h − 2 vijkl 2 aij (akl cp1 + a1kl ) Vp2 lnVp + 2bij (bkl cm1 + b1kl ) Vm2 lnVm

 aij (bkl cm1 + b1kl ) + bij (akl cp1 + a1kl ) Vp Vm (lnVp + lnVm ) − (aij Vp + bij Vm )      × akl (cp2 − cp1 ) + a1kl cp1 + a2kl Vp + bkl (cm2 − cm1 ) + b1kl cm1 + b2kl Vm

+

−



io  1 ,(B-11) (aij cp1 + a1ij )Vp + (bij cm1 + b1ij )Vm (akl c1p + a1kl )Vp + (bkl cm1 + b1kl )Vm 2

where i, j, k, l = 1, 2. Also b V2,old

  n 2    1 π2 = vijkl Vp aij + Vm bij Vp akl + Vm bkl 1 + + Vp Vm aij bkl lnVp − lnVm 2 8κ 6   o + 2 Vp aij lnVp + Vm bij lnVm Vp akl lnVp + Vm bkl lnVm . (B-12)

For the final “counter-term” diagram (c) we need to introduce the coefficients δvij , δuij , δwij whose values will be presented shortly (Appendix C). From eq.(11) δV1 = (δV1 )ij

=

λm 2 2 λσ 4 i 1 2ǫ h φ λφ 4 µ δ0 φ + δ0m φ σ + δ0σ σ , then εκ 4! 4 4! i 1 2ǫ h µ δvij + ǫ δuij + ǫ2 δwij , εκ

(B-13)

where (δV1 )ij = ∂ 2 (δV1 )/∂si ∂sj , i, j = 1, 2, s1 , s2 = φ, σ. With this notation, we find for diagram (c): h i h
1 n Vp aij [c1p − lnVp ] + a1ij + δu a V + b V δw ij ij p ij m ij 2 2κ h


i + Vm bij [c1m − lnVm ] + b1ij − δvij Vp lnVp aij c1p + a1ij + Vm lnVm bij c1m + b1ij

c V2,n =

and


io
− Vp aij c2p + [a1ij − aij ]c1p + a2ij − Vm bij c2m + [b1ij − bij ]c1m + b2ij

(B-14)

  h π2  π 2 i 1 2 2 ln V + 1 + ln V + 1 + + V b . V a δv p m m ij p ij ij 4κ2 6 6

(B-15)

c V2,old =

15

C

Coefficients entering the two-loop potential

Below we provide the expressions of the various coefficients introduced in the rhs of eq.(B-2), (B-13) and used in Appendix B. The coefficients vijkl , vijk are functions of λ’s and fields        

v1111 v1112 v1122 v1222 v2222

u1111 u1112 u1122 u1222 u2222

w1111 w1112 w1122 w1222 w2222

        =      

λφ 0 0 2λφ σφ 2 λm 3λm − λφ σφ2 φ φ3 2 0 3 λφ σ3 + 2λm σ 4 φ2 λσ − 21 λφ σφ4 + 25 6 λσ − λm σ2

0 2λφ σφ 2 λφ φσ2 + 5λm 3 8λm φσ − 43 λφ σφ3 φ2 φ4 4 3 λφ σ4 + 10λσ − 2λm σ2

       

(C-1)

and     

v111 v112 v122 v222

u111 u112 u122 u222

w111 w112 w122 w222

  λφ φ   λm σ   =    λm φ λσ σ

0 φ2 λφ σ + σλm 3 3λm φ − 13 λφ σφ2 φ4 φ2 1 13 6 λφ σ3 + 6 λσ σ + λm σ

0 φ2 λφ σ + λm σ φ3 1 3 λφ σ2 + 5λm φ 4 φ2 − 31 λφ σφ3 + 11 3 λσ σ + 4λm σ



  .  

(C-2)

Further, coefficients δvij , δuij and δwij of eq.(B-13) are δv11 = δv12 =


 1  3 λ2φ + λ2m φ2 + λm (λφ + 4λm + λσ ) σ 2 , 4κ 1 λm (λφ + 4λm + λσ ) φσ 2κ

δu12 = δw12 = δv22 = δu22 = δw22 =

δu11 = δw11 = 0

(C-3) (C-4)


φ 1  2 λφ + λ2m φ2 + λm (λφ + 4λm + λσ ) σ 2 2κ σ

(C-5)


2
2 1  2 2 2 4λ + λ λ + λ λ φ + 3 λ + λ σ m m σ φ m m σ 4κ2

(C-6)



 1 1  − λ2φ + λ2m φ4 + 6λm (λφ + 4λm + λσ ) φ2 σ 2 + 7 λ2σ + λ2m σ 4 (C-7) 2 2 8κ σ

 1 1  2 λφ + λ2m φ4 + 10λm (λφ + 4λm + λσ ) φ2 σ 2 + 9 λ2σ + λ2m σ 4 2 2 8κ σ

(C-8)

The coefficients c1p , c2p , c1m , c2m , also Vp and Vm introduced in eq.(B-2) have the expressions: "

Vp c1p Vm c1m

#

1 1 = 24 σ 2

"

−λφ φ4 + 18λm φ2 σ 2 + 7λσ σ 4 + R1 −λφ φ4 + 18λm φ2 σ 2 + 7λσ σ 4 − R1

16

#

(C-9)

where R1 = + and S2 =


1 h λφ (λφ − λm ) φ6 + 15λφ λm − λφ λσ + 18λ2m φ4 σ 2 2S i
−7λφ λσ + 78λ2m + 25λm λσ φ2 σ 4 + 7λσ (−λm + λσ ) σ 6

(C-10)

i
1h (λφ − λm )2 φ4 + 2 λφ λm + 7λ2m − λφ λσ + λm λσ φ2 σ 2 + (λm − λσ )2 σ 4 , (C-11) 4

while Vm and Vp (see also (B-3)) are "

Vp Vm

#

1 = 4

"

(λσ + λm ) σ 2 + (λφ + λm ) φ2 + 2S (λσ + λm ) σ 2 + (λφ + λm ) φ2 − 2S

#

.

(C-12)

For c2p , c2m (with the above Vp and Vm ): "

Vp c2p Vm c2m

#

1 1 = 24 σ 2

"

λφ φ4 + 30λm φ2 σ 2 + 9λσ σ 4 + R2 λφ φ4 + 30λm φ2 σ 2 + 9λσ σ 4 − R2

#

,

(C-13)

where
1 h 2 10 σ φ λφ 13λ3φ + 27λφ λ2m − 39λ2φ λm + 3λ3m + σ 4 φ8 − 23λσ λ3φ 2 3 σ S
+ 3λφ λ2m [3λσ + 16λφ ] + 5λ2φ λm [6λσ + 25λφ ] − 423λφ λ3m + 90λ4m

R2 =

+ σ 6 φ6 λσ λ2φ [7λσ − 27λφ ] + 3λ3m [99λσ + 151λφ ] + λφ λ2m [553λφ − 615λσ ]


+ λσ λφ λm [9λσ + 65λφ ] + 2034λ4m + σ 8 φ4 3λ2σ λφ [λσ + 27λφ ] + 3λ3m [641λσ + 261λφ ]


+ λσ λ2m [351λσ − 521λφ ] − λσ λφ λm [361λσ + 81λφ ] + 3438λ4m + σ 10 φ2 − 81λ3σ λφ + λσ λ2m [292λσ − 81λφ ] + 9λ2σ λm [19λσ + 18λφ ] − 609λσ λ3m + 342λ4m i 3 12 − 27σ λσ [λm − λσ ] .




(C-14)

Finally, aij , a1ij , a2ij , bij , b1ij , b2ij introduced in (B-2) and used in Appendix B have the values

a11 = 1 − a22 = b22 = 1 − b11 = a12 = a21 = −b12 = −b21 =

 1  1 + λφ φ2 + λm (−φ2 + σ 2 ) − λσ σ 2 (C-15) 2 4S

λm φ σ S

17

(C-16)


4 λm φ2 h φ − 2λ + 3λ λ m φ φ 6 S3
i
+ 2 λφ λσ − 4λφ λm − 6λ2m φ2 σ 2 − 6λ2m + λm λσ σ 4

a111 = −a122 = −b111 = b122 =

with S of eq.(C-11). Also

(C-17)


φ h 6 φ λφ 2λ2φ − 5λφ λm + 3λ2m 3 24σS
−4λσ λ2φ + 5λφ λm [λσ + 2λφ ] + λφ λ2m − 12λ3m

a112 = a121 = −b112 = −b121 = + φ4 σ 2

Further

+ φ2 σ 4 2λ2σ λφ + λ2m [14λφ − 13λσ ] − 9λσ λφ λm + 6λ3m
i + σ 6 λm −λ2σ − 5λσ λm + 6λ2m .

a211 = −a222 = −b211 = b222 =




(C-18)


φ2 n 10 2 φ λφ − 4λ3φ − 31λφ λ2m + 20λ2φ λm + 15λ3m 2 5 288σ S

i h + φ8 σ 2 λφ 12λσ λ3φ + λφ λ2m [31λσ + 180λφ ] − 20λ2φ λm [2λσ + 3λφ ] + 5λφ λ3m − 192λ4m 6 4

+ 2φ σ

h

− 6λ2σ λ3φ + λφ λ3m [44λφ − 163λσ ] − λ2φ λ2m [19λσ + 132λφ ]

h i + 2λσ λ2φ λm [5λσ + 28λφ ] + 450λφ λ4m + 144λ5m − 2φ4 σ 6 − 2λ3σ λ2φ + λ4m [270λφ −

96λσ ] + λφ λ3m [236λφ − 463λσ ] + λσ λφ λ2m [71λσ − 118λφ ] + 22λ2σ λ2φ λm + 1008λ5m

h + φ2 σ 8 λm − 8λ3σ λφ + 12λ3m [29λσ − 31λφ ] + λσ λ2m [184λφ − 117λσ ] + 49λ2σ λφ λm

i
o − 468λ4m + σ 10 3λ2m − 7λ3σ + 16λσ λ2m + 27λ2σ λm − 36λ3m .

(C-19)

Finally

a212 = a221 = −b212 = −b221 = −

h
2 φ − φ12 λ2φ (3λm − 2λφ ) λm − λφ 3 5 576σ S

+ 2σ 2 φ10 λφ − 3λ3φ [λσ + 2λφ ] − λφ λ2m [4λσ + 99λφ ] + λ2φ λm [7λσ + 57λφ ]


+ 22λφ λ3m + 30λ4m + σ 4 φ8 2λσ λ3φ [3λσ + 17λφ ] + λφ λ3m [160λσ + 301λφ ]

 
+ 2λ2φ λ2m [28λσ + 237λφ ] − λ2φ λm 184λσ λφ + 7λ2σ + 84λ2φ − 972λφ λ4m − 72λ5m + 2σ 6 φ6 − λ2σ λ2φ [λσ + 15λφ ] − 6λ4m [7λσ − 68λφ ] + λφ λ3m [379λφ − 651λσ ]  
+λφ λ2m −5λσ λφ + 71λ2σ − 108λ2φ + λσ λ2φ λm [13λσ + 75λφ ] + 1116λ5m 18

i

  + σ 8 φ4 6λ3σ λ2φ + 12λ4m [92λσ + 45λφ ] + λ3m 540λσ λφ + 59λ2σ − 264λ2φ −6λσ λφ λ2m [91λσ − 41λφ ] + λ2σ λφ λm [44λσ − 37λφ ] + 324λ5m

+ 2σ 10 φ2 λ4σ λφ + 6λ4m [31λσ − 13λφ ] + λσ λ3m [89λφ − 103λσ ] +λ2σ λ2m [41λσ + 8λφ ] − 20λ3σ λφ λm + 72λ5m




+ σ 12 (−λm ) (λm − λσ )2 −11λ2σ + 24λσ λm + 36λ2m


i

,




(C-20)

which enter in the expression of the two-loop potential.

Acknowledgements: The work of D. Ghilencea was supported by a grant from Romanian National Authority for Scientific Research (CNCS-UEFISCDI) under project number PNII-ID-PCE-2011-3-0607. The work of P. Olszewski and Z. Lalak was supported by the Polish NCN grants DEC-2012/04/A/ST2/00099 and 2014/13/N/ST2/02712.

References [1] For an early work see C. Wetterich, “Cosmology and the Fate of Dilatation Symmetry,” Nucl. Phys. B 302 (1988) 668. For the local conformal symmetry, see I. Bars, P. Steinhardt and N. Turok, “Local Conformal Symmetry in Physics and Cosmology,” Phys. Rev. D 89 (2014) no.4, 043515 [arXiv:1307.1848 [hep-th]] and references therein. [2] F. Englert, C. Truffin and R. Gastmans, “Conformal Invariance in Quantum Gravity,” Nucl. Phys. B 117 (1976) 407. S. Deser, “Scale invariance and gravitational coupling,” Annals Phys. 59 (1970) 248. [3] M. Shaposhnikov and D. Zenhausern, “Quantum scale invariance, cosmological constant and hierarchy problem,” Phys. Lett. B 671 (2009) 162 [arXiv:0809.3406 [hep-th]]. [4] M. E. Shaposhnikov and F. V. Tkachov, “Quantum scale-invariant models as effective field theories,” arXiv:0905.4857 [hep-th]. [5] R. Armillis, A. Monin and M. Shaposhnikov, “Spontaneously Broken Conformal Symmetry: Dealing with the Trace Anomaly,” JHEP 1310 (2013) 030 [arXiv:1302.5619 [hep-th]]. [6] D. M. Ghilencea, “Manifestly scale-invariant regularization and quantum effective operators,” Phys. Rev. D 93 (2016) no.10, 105006 [arXiv:1508.00595 [hep-ph]]. [7] D. M. Ghilencea, “One-loop potential with scale invariance and effective operators”, arXiv:1508.00595 [hep-ph], Proceedings, 15th Hellenic School and Workshops on Elementary Particle Physics and Gravity (CORFU2015) : Corfu, Greece, Sep 1-25, 2015 19

[8] C. Tamarit, “Running couplings with a vanishing scale anomaly,” JHEP 1312 (2013) 098 [arXiv:1309.0913 [hep-th]]. [9] W. A. Bardeen, “On naturalness in the standard model,” FERMILAB-CONF-95-391T, C95-08-27.3. [10] P. G. Ferreira, C. T. Hill and G. G. Ross, “Scale-Independent Inflation and Hierarchy Generation,” [arXiv:1603.05983 [hep-th]]. [11] M. Shaposhnikov and D. Zenhausern, “Scale invariance, unimodular gravity and dark energy,” Phys. Lett. B 671 (2009) 187 [arXiv:0809.3395 [hep-th]]. [12] I. Oda, “Higgs Mechanism in Scale-Invariant Gravity,” Adv. Stud. Theor. Phys. 8 (2014) 215 [arXiv:1308.4428 [hep-ph]]. I. Oda, “Higgs Mechanism in Scale-Invariant Gravity,” Adv. Stud. Theor. Phys. 8 (2014) 215 [arXiv:1308.4428 [hep-ph]]. [13] A. Kobakhidze, “Quantum relaxation of the Higgs mass,” Eur. Phys. J. C 75 (2015) no.8, 384 [arXiv:1506.04840 [hep-ph]]. [14] K. Allison, C. T. Hill and G. G. Ross, “Ultra-weak sector, Higgs boson mass, and the dilaton,” Phys. Lett. B 738 (2014) 191 [arXiv:1404.6268 [hep-ph]]. [15] R. Foot, A. Kobakhidze, K. L. McDonald and R. R. Volkas, “Poincar´e protection for a natural electroweak scale,” Phys. Rev. D 89 (2014) no.11, 115018 [arXiv:1310.0223 [hep-ph]]. See also related [16]. [16] R. Foot, A. Kobakhidze, K. L. McDonald and R. R. Volkas, “A Solution to the hierarchy problem from an almost decoupled hidden sector within a classically scale invariant theory,” Phys. Rev. D 77 (2008) 035006 [arXiv:0709.2750 [hep-ph]]. [17] A. I. Davydychev and J. B. Tausk, “Two loop selfenergy diagrams with different masses and the momentum expansion,” Nucl. Phys. B 397 (1993) 123. [18] C. Ford, I. Jack and D. R. T. Jones, “The Standard model effective potential at two loops,” Nucl. Phys. B 387 (1992) 373 Erratum: [Nucl. Phys. B 504 (1997) 551] [hep-ph/0111190]. [19] J. R. Espinosa, M. Garny and T. Konstandin, “Interplay of Infrared Divergences and Gauge-Dependence of the Effective Potential,” arXiv:1607.08432 [hep-ph]. J. EliasMiro, J. R. Espinosa and T. Konstandin, “Taming Infrared Divergences in the Effective Potential,” JHEP 1408 (2014) 034 [arXiv:1406.2652 [hep-ph]]. S. P. Martin, “Taming the Goldstone contributions to the effective potential,” Phys. Rev. D 90 (2014) no.1, 016013 [arXiv:1406.2355 [hep-ph]].

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